The First-Order Syntax of Variadic Functions

Alexander, S. A.

doi:10.1215/00294527-1731380

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An Abstract Chain Rule1

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2022

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(2 citation statements)

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“…For any assignment s, if φ(s) is as in (1), then T φ(s) = U φ(s) by Definition 5.7. Thus φ(T ) s = φ(U ) s by (1). By arbitrariness of s, φ(T ) ≡ φ(U ).…”

Section: An Abstract Chain Rulementioning

confidence: 94%

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Formal differential variables and an abstract chain rule

Alexander¹

2022

Preprint

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One shortcoming of the chain rule is that it does not iterate: it gives the derivative of f (g(x)), but not (directly) the second or higher-order derivatives. We present iterated differentials and a version of the multivariable chain rule which iterates to any desired level of derivative. We first present this material informally, and later discuss how to make it rigorous (a discussion which touches on formal foundations of calculus). We also suggest a finite calculus chain rule (contrary to Graham, Knuth and Patashnik's claim that "there's no corresponding chain rule of finite calculus").

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“…For any assignment s, if φ(s) is as in (1), then T φ(s) = U φ(s) by Definition 5.7. Thus φ(T ) s = φ(U ) s by (1). By arbitrariness of s, φ(T ) ≡ φ(U ).…”

Section: An Abstract Chain Rulementioning

confidence: 94%

“…It is also possible to incorporate summation notation formally into Definition 5.3, but the details are complicated. See[1].…”

mentioning

confidence: 99%